The present invention relates to methods and systems for accelerating the computation of L* for magnetospheric research and applications. In particular, it relates to accelerating such calculations by using a neural network as a surrogate model for a physics-based model of calculating L*.
The atmosphere is defined by the air surrounding our planet, the magnetosphere is the area around our planet that is defined by the magnetic field of the Earth. FIG. 1 illustrates the pattern of the radiation belts around the Earth. The magnetosphere was discovered by artificial satellite activity, in particular by the activities of the artificial satellite Explorer 1 in 1958. For the remainder of the application the term satellite should be read to mean man-made artificial satellites, and not natural satellites, such as the moon, orbiting Earth.
The effect of the magnetosphere on satellites has been an important consideration in the design and operation of satellites. Satellites themselves are important for a wide range of activities including: defense, astronomy, biological experimentation, communications, navigation, reconnaissance, Earth observation, manned space activities (space stations), and weather observation. Today there are about 3,000 useful satellites in orbit.
For example, DirecTV has contracted with Boeing for the construction, launch and insurance of three HDTV communication satellites for $300 million per satellite. Clearly, the investments in these satellites are very serious economic investments for private enterprises. Public investment is much larger. According to U.S. News and World Report, the United States had invested over $200 billion in spy satellites as of 2003. The national security interests of the United States as of 1999 were summarized in a report by National Security Space Architect. (Ref. 4) (References referred to as “Ref. <number>”)
A report by the reinsurance company, Swiss R E, Space Weather: Hazard to the Earth? reports that space weather is also believed to be a present risks to terrestrial electronics, space flight, aviation, telecommunications, electric power transmission, the oil and gas industry and railways. The damage done by a specific event is chronicled in The Halloween Space Weather Storms of 2003 by the National Oceanic and Atmospheric Administration (NOAA). (Ref. 5) Among the events chronicled by NOAA was the disabling of a $640 million dollar Japanese satellite to monitor climate change.
One major risk to the satellites is charged particles. The sun sends electrically charged particles (electrons and protons) to the Earth with a dynamic solar wind. The magnetic field in the solar wind will interact with the magnetic field of the Earth by reconnection of field lines which allows solar wind particles to enter the magnetosphere. Subsequently particles inside the magnetosphere are accelerated by wave-particle interactions creating the highly energetic radiation belts. Presently there are about 6,000 failed satellites
FIG. 2 illustrates how these particles become trapped along a magnetic field line. The particle then follows a path near the magnetic field line, confined in a cylinder wrapped around that field line called a flux tube. When the flux tube leads the charged particles near the Earth they reach a mirror point. At that point the charged particle reflects, and travels along the flux tube until it reaches the mirror point at the other end of the magnetic field line.
These trapped electrically charged particles can harm satellites in a number of ways. For example:                1) protons (positive charges) can damage the solar panels that are used to power satellites;        2) electrons can get into computer chips and change programs and data; and        3) electrical charge can build up on the interior or exterior of the satellite, discharge like a small lightning bolt, and damage the satellite.        
Satellites are designed with these risks in mind. However, the shielding material is typically very heavy. Therefore increasing the shielding also increases the cost of constructing, launching and deploying the satellite. While understanding the effect of the magnetosphere on these satellites is important, achieving that understanding is difficult. Also, because of the highly variable nature of space weather, it may be preferable to design the satellite to operate up to a threshold of magnetospheric disturbance, but plan to put the satellite in to a protected mode, such as by orienting the satellite or turning off the satellite, in order to preserve the satellite in weather conditions above the threshold of magnetospheric disturbance.
Space weather modeling, forecasts, and predictions require detailed information about the Earth's magnetic field. In radiation belt research many results depend on the magnetic field and the particle drift shells illustrated in FIGS. 1 and 2. One of the most important parameters is called L* which is the magnetic drift invariant, and is one of the three adiabatic invariants along with μ and K. L* is defined as:
      L    *    =            2      ⁢              πμ        0                    Φ      ⁢                          ⁢              R        E            Where μ0 is Earth's magnetic moment, Φ as the flux enclosed by the drift shell of a particle on a given field line as illustrated by FIGS. 2 and 9, and the Earth radius RE. In a static dipole field, Φ can be calculated analytically whereas Φ requires detailed and time consuming numerical integration in a more realistic magnetic field.
The magnetic fields of Earth can be modeled at different levels of detail. For example, a very simple model is that of the Earth as simple magnet of unvarying magnetic field. More sophisticated models account for the shape of the magnetic field, and the location of magnetic field lines being dynamic and shaped by events such as changes in the solar wind.
Currently many empirical magnetic field models exist but it can take a long time to calculate L* using more sophisticated models. (Ref. 6) Where simple models from the 1970s or 1980s can perform a calculation in 3-6 minutes for one day's worth of data in one minute resolution, the models of the 1990s or 2000s take 1.5 to 4.5 hours, depending upon their sophistication. The performance of several different magnetic field models has been recently studied and found to be wanting. (Ref. 6)
Because of these long computing times, workers in the field often pick simplistic models over more accurate ones risking strong inaccuracies. (Ref. 6) Huang et al. quantified recently the effect of choosing a magnetic field model for radiation belt studies and concludes that the global inaccuracies of magnetic field models could alter the results of the inferred radial profiles of phase space densities of radiation belt electrons. (Ref. 7) Huang et al. also found that during quiet times the resulting values from the L□ calculation between models can vary by 13% and during storm times up to 50%. (private communications). The use of these simple models naturally results in either less accurate assumptions about the magnetospheric environment or calculations that cannot be used on practical time scales to make decisions informed by the magnetospheric environment.
Accordingly, there is a long-felt need for the ability to apply high-quality physics-based models of magnetospheric space weather on practical timescales. Rapid access to results from the best models would result in better research, better understanding of space weather, better design of satellites, and an improved ability to operate a satellite relative to magnetospheric conditions and its design.    (Ref. 4) United States Air Force Brigadier General Howard J. Mitchell in Space Weather Architecture Final Report (Mar. 22, 1999)    (Ref. 5) NOAA Technical Memorandum OAR SEC-88 (June 2004)    (Ref. 6) McCollough, J. P., J. L. Gannon, D. N. Baker, and M. Gehmeyr (Ref. 2008), A statistical comparison of commonly used external magnetic field models, Space Weather, 6, S10001, doi:10.1029/2008SWO00391.    (Ref. 7) Huang, C.-L., H. E. Spence, H. J. Singer, and N. A. Tsyganenko (Ref. 2008), A quantitative assessment of empirical magnetic field models at geosynchronous orbit during magnetic storms, Journal of Geophysical Research (Space Physics), 113, 04, 208.    (Ref. 8) Tsyganenko, N. A. (Ref. 2002a), A model of the near magnetosphere with a dawn-dusk asymmetry 1. mathematical structure, Journal of Geophysical Research (Space Physics), 107, 1179.    (Ref. 9) Tsyganenko, N. A. (Ref. 2002b), A model of the near magnetosphere with a dawn-dusk asymmetry 2. parameterization and fitting to observations, Journal of Geophysical Research (Space Physics), 107, 1176.    (Ref. 10) Tsyganenko, N. A., and M. I. Sitnov (Ref. 2005), Modeling the dynamics of the inner magnetosphere during strong geomagnetic storms, Journal of Geophysical Research (Space Physics), 110, 03, 208.    (Ref. 11) Myers, R. H., and D. C. Montgomery (Ref. 2002), Response surface methodology process and product optimization using designed experiments, Wiley, New York; Chichester.    (Ref. 12) Kleijnen, J. P. C. (Ref. 2008), Design and analysis of simulation experiments, International series in operations research and management science, 111, Springer, New York.    (Ref. 13) Barron, A. (Ref 1991), Approximation bounds for superpositions of a sigmoidal function, Information Theory, IEEE Transactions on, pp. 85-85. Barron, A. (Ref. 1993), Universal approximation bounds for superpositions of a sigmoidal function, Information Theory, IEEE Transactions on, 39 (Ref. 3), 930-945. Barron, A. R. (Ref. 1994), Approximation and estimation bounds for artificial neural networks, Machine Learning, 14    (Ref. 14) Cybenko, G. (Ref. 1989), Approximation by superpositions of a sigmoidal function, Mathematics of control, signals, and systems, 2 (Ref. 4), 303.    (Ref. 15) Reed, R. D., and R. J. Marks (Ref. 1999), Neural smithing: supervised learning in feedforward artificial neural networks, The MIT Press, Cambridge, Mass.    (Ref. 16) Miller, P. (Ref. 2008), pympi: Putting the py in mpi, http://pympi.sourceforge.net/.    (Ref. 17) Nash, S. (Ref. 1984), Newton-type minimization via the lanczos method, SIAM Journal (Ref. 4) on Numerical Analysis, 21 (Ref 4), 770-788.    (Ref. 18) Peterson, P. (Ref 2007), F2py: Fortran to python interface generator.    (Ref. 19) Tsyganenko, N. A., H. J. Singer, and J. C. Kasper (Ref. 2003), Storm-time distortion of the inner magnetosphere: How severe can it get?, Journal of Geophysical Research (Space Physics), 108, 1209.    (Ref. 20) Wojciechowski, M. (Ref 2007), ffnet: Feed-forward neural network for python, http://ffnet.sourceforge.net/    (Ref. 21) J. G. Roederer “Dynamics of Geomagnetically Trapped Radiation”, Springer Verlag, 1970    (Ref 22) Boscher, D., S. Bourdarie, P. O'Brien, and T. Guild (Ref. 2007), ONERA-DESP library v4.1, http://craterre.onecert.fr/support/user guide.html.